Clustering and classification
date()## [1] "Fri Nov 18 19:49:19 2022"# LOADING THE DATA
# access the MASS package
library(MASS)
# load the data
data("Boston")
# define subset of data for boxplots
box <- dplyr::select(Boston, -tax, -black)
# explore the dataset
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00# plot distributions and matrix of the variables
boxplot(box)
boxplot(Boston$tax)
boxplot(Boston$blac)
p <- pairs(Boston)
p## NULLThe Boston data contain information from 506 towns on factors such as crime rate, residential land, nitrogen oxides concentration, rooms per dwelling, tax rate, pupil-teacher ratio, population status and value of owner-occupied homes. Altogether 14 variables are included. No values are missing.
Variable distributions are mostly skewed. Median age of population is 77.5 years (range 2.9 to 100 years). Crime rate varies from 0.006 to 89.0, tax rate from 187 to 711 per 10,000 $, pupil-teacher ratio from 12.6 to 22.0, proportion of lower population status from 1.7% to 38.0% and nitrogen oxides concentration 0.39 to 0.87 parts per 10 million between towns.
# EXPLORING THE DATA
# access the tidyr and corrplot libraries
library(tidyr)
library(corrplot)## corrplot 0.92 loaded# calculate the correlation matrix and round it
cor_matrix <- cor(Boston) %>% round(digits = 2)
# print the correlation matrix
cor_matrix ## crim zn indus chas nox rm age dis rad tax ptratio
## crim 1.00 -0.20 0.41 -0.06 0.42 -0.22 0.35 -0.38 0.63 0.58 0.29
## zn -0.20 1.00 -0.53 -0.04 -0.52 0.31 -0.57 0.66 -0.31 -0.31 -0.39
## indus 0.41 -0.53 1.00 0.06 0.76 -0.39 0.64 -0.71 0.60 0.72 0.38
## chas -0.06 -0.04 0.06 1.00 0.09 0.09 0.09 -0.10 -0.01 -0.04 -0.12
## nox 0.42 -0.52 0.76 0.09 1.00 -0.30 0.73 -0.77 0.61 0.67 0.19
## rm -0.22 0.31 -0.39 0.09 -0.30 1.00 -0.24 0.21 -0.21 -0.29 -0.36
## age 0.35 -0.57 0.64 0.09 0.73 -0.24 1.00 -0.75 0.46 0.51 0.26
## dis -0.38 0.66 -0.71 -0.10 -0.77 0.21 -0.75 1.00 -0.49 -0.53 -0.23
## rad 0.63 -0.31 0.60 -0.01 0.61 -0.21 0.46 -0.49 1.00 0.91 0.46
## tax 0.58 -0.31 0.72 -0.04 0.67 -0.29 0.51 -0.53 0.91 1.00 0.46
## ptratio 0.29 -0.39 0.38 -0.12 0.19 -0.36 0.26 -0.23 0.46 0.46 1.00
## black -0.39 0.18 -0.36 0.05 -0.38 0.13 -0.27 0.29 -0.44 -0.44 -0.18
## lstat 0.46 -0.41 0.60 -0.05 0.59 -0.61 0.60 -0.50 0.49 0.54 0.37
## medv -0.39 0.36 -0.48 0.18 -0.43 0.70 -0.38 0.25 -0.38 -0.47 -0.51
## black lstat medv
## crim -0.39 0.46 -0.39
## zn 0.18 -0.41 0.36
## indus -0.36 0.60 -0.48
## chas 0.05 -0.05 0.18
## nox -0.38 0.59 -0.43
## rm 0.13 -0.61 0.70
## age -0.27 0.60 -0.38
## dis 0.29 -0.50 0.25
## rad -0.44 0.49 -0.38
## tax -0.44 0.54 -0.47
## ptratio -0.18 0.37 -0.51
## black 1.00 -0.37 0.33
## lstat -0.37 1.00 -0.74
## medv 0.33 -0.74 1.00# visualize the correlation matrix
corrplot(cor_matrix, method="circle", type = "upper", tl.pos = "d", tl.cex = 0.6)
The variables are mostly correlated with each other, except for the
Charles River dummy variable describing if the tract bounds river or not
which is correlated only mildly with the median value of owner-occupied
homes (rho = 0.18). There are remarkably high correlations between age
and nitrogen oxides concentration (rho = 0.73) and mean distance to main
employment centres (rho = -0.75), between non-retail business acres and
nitrogen oxides concentration (rho = 0.76), mean distance to main
employment centres (rho = -0.71) and tax rate (rho = 0.72), between
average number of rooms per dwelling and median value of owner-occupied
homes (rho = 0.70), between accessibility to radial highways and tax
rate (rho = 0.91), and between proportion of lower population status and
median value of owner-occupied homes (rho = -0.74).
# SCALING THE DATASET AND CREATING 'CRIME' VARIABLE
# accessing library dplyr
library(dplyr)##
## Attaching package: 'dplyr'## The following object is masked from 'package:MASS':
##
## select## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union# center and standardize variables
boston_scaled <- scale(Boston)
# summaries of the scaled variables
summary(boston_scaled)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865# plot distributions
boxplot(boston_scaled)
# class of the boston_scaled object
class(boston_scaled)## [1] "matrix" "array"# change the object to data frame
boston_scaled <- scale(Boston) %>% as.data.frame
# summary of the scaled crime rate
summary(boston_scaled$crim)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.419367 -0.410563 -0.390280 0.000000 0.007389 9.924110# create a quantile vector of crim and print it
bins <- quantile(boston_scaled$crim)
bins## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610# create a categorical variable 'crime'
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE)
# look at the table of the new factor crime
table(crime)## crime
## [-0.419,-0.411] (-0.411,-0.39] (-0.39,0.00739] (0.00739,9.92]
## 127 126 126 127# remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)
# add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)Scaling removes most of the skewness in variables.
# LINEAR DISCRIMINANT ANALYSIS
# number of rows in the Boston dataset
n <- nrow(boston_scaled)
# choose randomly 80% of the rows
ind <- sample(n, size = n * 0.8)
# create train set
train <- boston_scaled[ind,]
# create test set
test <- boston_scaled[-ind,]
# save the correct classes from test data
correct_classes <- test$crime
# remove the crime variable from test data
test <- dplyr::select(test, -crime)
# linear discriminant analysis
lda.fit <- lda(crime ~ ., data = train)
# print the lda.fit object
lda.fit## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## [-0.419,-0.411] (-0.411,-0.39] (-0.39,0.00739] (0.00739,9.92]
## 0.2351485 0.2549505 0.2623762 0.2475248
##
## Group means:
## zn indus chas nox rm
## [-0.419,-0.411] 1.03805753 -0.8673084 -0.065113266 -0.8969741 0.51870329
## (-0.411,-0.39] -0.08407022 -0.3258744 -0.004759149 -0.5921620 -0.13484041
## (-0.39,0.00739] -0.39744090 0.2785509 0.247665303 0.4910288 0.07048788
## (0.00739,9.92] -0.48724019 1.0171519 -0.036103054 1.0451102 -0.42298198
## age dis rad tax ptratio
## [-0.419,-0.411] -0.8913572 0.8321081 -0.6953589 -0.7215359 -0.55951665
## (-0.411,-0.39] -0.3518447 0.3998278 -0.5425547 -0.5271189 -0.02609972
## (-0.39,0.00739] 0.4881596 -0.4348846 -0.4163054 -0.2747918 -0.34244864
## (0.00739,9.92] 0.8035884 -0.8508245 1.6377820 1.5138081 0.78037363
## black lstat medv
## [-0.419,-0.411] 0.38495934 -0.78887138 0.601314448
## (-0.411,-0.39] 0.31628070 -0.14223160 0.001077744
## (-0.39,0.00739] 0.05480403 0.08997301 0.143115732
## (0.00739,9.92] -0.81955028 0.85849096 -0.659643108
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.10435234 0.77940001 -1.04955970
## indus 0.02755964 -0.10018845 0.16285258
## chas -0.08663762 -0.05477837 0.06882547
## nox 0.38595980 -0.85721244 -0.99483548
## rm -0.09419823 -0.09116794 -0.21282756
## age 0.24678330 -0.37120746 -0.06775274
## dis -0.08820886 -0.42222933 0.45816865
## rad 3.15999132 1.09317664 0.11783354
## tax 0.03376795 -0.23043515 0.20600241
## ptratio 0.15123659 -0.01128501 -0.10625810
## black -0.10344385 -0.01056462 0.06821921
## lstat 0.24365608 -0.25977556 0.27634945
## medv 0.23513551 -0.42462104 -0.13771239
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9404 0.0459 0.0137# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(train$crime)
# plot the lda results
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 2)
In linear discriminant analysis, the highest crime class almost solely
occupies one cluster and the other crime classes are mixed in the other
cluster. The most influential line separators seem to be accessibility
to radial highways, proportion of residential land zoned for lots over
2323 m2 and nitrogen oxides concentration, most likely implying
differences between rural and urban environments. The first discriminant
function separates 94.7% of the population.
# PREDICTING WITH THE LDA MODEL
# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class) %>% addmargins## predicted
## correct [-0.419,-0.411] (-0.411,-0.39] (-0.39,0.00739] (0.00739,9.92]
## [-0.419,-0.411] 15 16 1 0
## (-0.411,-0.39] 3 15 5 0
## (-0.39,0.00739] 1 9 9 1
## (0.00739,9.92] 0 0 0 27
## Sum 19 40 15 28
## predicted
## correct Sum
## [-0.419,-0.411] 32
## (-0.411,-0.39] 23
## (-0.39,0.00739] 20
## (0.00739,9.92] 27
## Sum 102The LDA model works reasonably well: it correctly predicts the crime class in 69 (68%) towns.
# DISTANCES AND CLUSTERING
# access library ggplot2
library(ggplot2)
# scale dataset
boston_scaled2 <- data.frame(scale(Boston))
# euclidean distance matrix
dist_eu <- dist(boston_scaled2)
# look at the summary of the distances
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970# manhattan distance matrix
dist_man <- dist(boston_scaled2, method = "manhattan")
# look at the summary of the distances
summary(dist_man)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.2662 8.4832 12.6090 13.5488 17.7568 48.8618# k-means clustering
km <- kmeans(boston_scaled2, centers = 3)
# plot the Boston dataset with clusters
pairs(boston_scaled2, col = km$cluster)
pairs(boston_scaled2[1:7], col = km$cluster)
pairs(boston_scaled2[8:14], col = km$cluster)
set.seed(123)
# determine the number of clusters
k_max <- 10
# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled2, k)$tot.withinss})
# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')
# k-means clustering
km <- kmeans(boston_scaled2, centers = 2)
# plot the Boston dataset with clusters
pairs(boston_scaled2, col = km$cluster)
pairs(boston_scaled2[1:7], col = km$cluster)
pairs(boston_scaled2[8:14], col = km$cluster)
Scaling reduces the distances and skewness of their distribution. For
k-means clustering, 2 centers seem to be the best choice. One cluster
seem to be associated with low crime rate, low proportion of non-retail
business acres, low nitrogen oxides concentration, younger age, high
mean distance to main employment centres, low accessibility to radial
highways, low tax rate, high difference in race proportions and high
median value of owner-occupied homes. This may refer to that clustering
identifies rural areas or otherwise less densely populated areas
vs. urban areas.
# BONUS SECTION: MORE ON K-MEANS CLUSTERING AND LINEAR DISCRIMINANT ANALYSIS
# k-means clustering
km2 <- kmeans(boston_scaled2, centers = 6)
# plot the Boston dataset with clusters
pairs(boston_scaled2, col = km2$cluster)
pairs(boston_scaled2[1:7], col = km2$cluster)
pairs(boston_scaled2[8:14], col = km2$cluster)
# linear discriminant analysis
lda.fit2 <- lda(km2$cluster ~ ., data = boston_scaled2)
# print the lda.fit object
lda.fit2## Call:
## lda(km2$cluster ~ ., data = boston_scaled2)
##
## Prior probabilities of groups:
## 1 2 3 4 5 6
## 0.06719368 0.12252964 0.24505929 0.18972332 0.29051383 0.08498024
##
## Group means:
## crim zn indus chas nox rm age
## 1 -0.1985497 -0.2602436 0.2799956 3.6647712 0.3830784 0.2756681 0.3721322
## 2 -0.4141953 2.3322773 -1.1788641 -0.2088275 -1.1887944 0.7188485 -1.4216721
## 3 1.1156495 -0.4872402 1.0149946 -0.2723291 0.9916473 -0.4276828 0.7515952
## 4 -0.3269211 -0.4816572 0.6406213 -0.2723291 0.4664828 -0.5082323 0.7668344
## 5 -0.3977957 -0.1563279 -0.5990216 -0.2723291 -0.6696819 -0.1686625 -0.6751063
## 6 -0.3732407 -0.1422288 -0.8310017 -0.2723291 -0.2005297 1.6901170 0.1841367
## dis rad tax ptratio black lstat
## 1 -0.4033382 0.001081444 -0.0975633 -0.39245849 0.17154271 -0.1643525
## 2 1.6223203 -0.666968948 -0.5535972 -0.82951564 0.35193833 -0.9665363
## 3 -0.8170870 1.659602895 1.5294129 0.80577843 -0.81154619 0.9129958
## 4 -0.5732656 -0.602637816 -0.1649468 0.21059409 0.04824716 0.5397714
## 5 0.5672190 -0.575610755 -0.6877857 -0.05330275 0.36267528 -0.3956902
## 6 -0.3232389 -0.511800977 -0.8155263 -1.10522083 0.34963036 -0.9616241
## medv
## 1 0.573340910
## 2 0.799806470
## 3 -0.771340259
## 4 -0.505310108
## 5 0.007601824
## 6 1.719927960
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3 LD4 LD5
## crim 0.034508803 0.05797085 -0.16103219 0.152322830 0.01702637
## zn 0.263383770 -0.23309900 -1.54525759 -1.099430119 0.78886901
## indus -0.073596625 0.35511069 0.27621202 -0.741302164 -0.03049072
## chas -5.833329077 -0.28810784 -0.25366217 -0.153849858 -0.20101832
## nox 0.012019761 -0.33200265 0.08267993 -0.138256930 0.38821434
## rm 0.077523647 0.02092389 -0.05279584 0.346603115 0.54639625
## age -0.078503080 0.10642631 0.50741330 -0.163867301 0.79431423
## dis -0.055475885 -0.36793105 -0.33307268 -0.006313437 -0.37843684
## rad -0.319867249 3.41804068 -1.06679562 1.629076215 -0.94341422
## tax 0.011360375 -0.30248228 -0.48448257 -0.924010759 0.59988910
## ptratio -0.010141142 -0.03367258 0.16740977 -0.504498820 0.12595753
## black -0.007393166 -0.08886704 0.01924355 0.045161817 -0.07202500
## lstat 0.123902259 0.13351602 0.01047765 0.007966329 0.40940453
## medv 0.203273034 -0.38745314 -0.07192725 0.516887596 0.76079378
##
## Proportion of trace:
## LD1 LD2 LD3 LD4 LD5
## 0.6292 0.2589 0.0730 0.0230 0.0159# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# plot the lda results
plot(lda.fit2, dimen = 2, col = km2$cluster, pch = km2$cluster)
lda.arrows(lda.fit2, myscale = 2)
Linear discriminant analysis produces three clusters, in which one
cluster is occupied by k-means cluster 1, another by k-means cluster 3,
and the third cluster contains the rest k-means clusters. The most
influential line separators seem to be accessibility to radial highways
and Charles River dummy variable. The first discriminant function
separates 62.9% and the second one 25.9% of the towns.
# SUPER-BONUS SECTION: 3D PLOTTING
# access library plotly
library(plotly)##
## Attaching package: 'plotly'## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layout# define a new object
model_predictors <- dplyr::select(train, -crime)
# check the dimensions
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
# k-means clustering
km3 <- kmeans(matrix_product, centers = 2)
km4 <- km3$cluster
# Create 3D plot
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers')plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = train$crime)plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = km4)The first matrix plot shows two clusters that are well separated from each other. In the second plot, one cluster is mainly occupied by the highest crime class. In the third plot, the clusters coincide fully with the k-means clustering with two centers.